Nonlinear Sampled-data Control Systems Research Articles

Partial decoupling of multivariable non–linear sampled–data control systems and the asymptotic stability analysis

The main purpose of this paper is to present some results concerning the stabilization of nonlinear sampled–data control systems. The stabilization is investigated by means of a particular decomposition of the state space, based on some state variable feedback, into two sub–spaces, one of them being an invariant sub– space.

Non-linear sampled-data control systems with arbitrary values of transportation lag

The effect of inherent finite delay which is not an integral multiple of the sampling period on the character of limit cycles in a second-order feedback control system with a relay type non-linearity is analysed. It is found that the addition of transportation log alters the character of the modes sustained by the system. All the modes that can be sustained for each fractional value of the delay which lies between zero and one and two and three sampling periods are determined by the discrete describing function method. In this connection the critical regions developed by the authors for periods ranging from 9T to 20T have been found to be very useful. The different sots of initial conditions which give rise to the various modes are also established via the state transition technique. It is observed that an increase in the transportation lag usually results in an increase in the amplitude as well as the period of oscillation. Square-wave dither is found to produce a large steady-state error for a step inpu...

Evaluation of response of nonlinear discrete systems using multidimensional Z transforms

Using the concept of a multidimensional Z transform, a method for solving time-invariant nonlinear difference equations is proposed. The solution is obtained in the form of a discrete Volterra series. Using the new technique, the response of a nonlinear sampled-data control system is obtained which compares very favourably with that obtained by other, more tedious and cumbersome, methods.

Transient response of a class of nonlinear sampled-data control systems

The particular form of the mathematical model which describes nonlinear sampled-data control systems gives a method of studying transient response of this process.

BIBO stability of nonlinear sampled-data multivariable feedback control systems

The definition of a state vector which takes into account the error functions of a nonlinear discrete multivariable plant allows one to obtain an estimate of the forced motions by utilising a characteristic property of the autonomous system. The result generalises that of Iwens and Bergen* obtained for monovariable plants via the discrete analogue of Popov's global-stability criterion.

Stabilization of non-linear sampled-data control systems with inherent hidden oscillations†

A relay introduced in the forward path of a sampled-data system deteriorates the intersample response of a linear system exhibiting hidden oscillations. It is established that sinusoidal dither has very fine control over the intersample response as well. The dither signal, incidentally, can be obtained by the addition of a minor loop around the relay. The zero-order hold is shown to be effective in eliminating the intersample ripples.

Fundamentals of non-linear control systems with pulse-frequency and pulse-width modulation

We consider non-linear sampled-data control systems consisting of a continuous linear part and a non-linear pulse modulator ( PM). The PM modulates a sequence of rectangular pulses by sign, frequency and duration, as a function of a linear combination of discrete values of the system coordinates. For the stated systems the concepts of the natural phase space E m with coordinates x n , x 1 n , …, x n ( m−1) and of the difference phase space D m with coordinates x n , x n+1 , …, x n+ m−1 are used. Non-linear difference equations of motion have been found in both the phase spaces, and the relation between them has been established. A sufficient condition for the asymptotic stability in-the-large of the systems being investigated is obtained on the basis of discrete analogs of the Lyapunov and LaSalle theorems. The condition has the form of a transcendental inequality, uniquely determined by the system parameters and by the Lyapunov function chosen. The problems of absolute stability and of the region of asymptotic stability also are solved with the aid of the inequality obtained. A procedure for determining the steady-state parameters of the system tracking a ramp input is given. For this steady-state mode sufficient conditions are obtained for stability in-the-large on the basis of Lyapunov's direct method. A discrete analog of the Yoshizawa theorem for the limit boundedness of non-linear sampled-data systems is formulated. On the basis of this theorem a procedure is proposed for determining the boundaries of the asymptotically stable set inside which all the phase trajectories of the system terminate.

An extended frequency-domain stability criterion for a class of sampled-data systems

A new frequency-domain stability criterion employing a Popov-type multiplier is obtained for a class of nonlinear sampled-data control systems containing a monotonically nondecreasing nonlinearity. This criterion is an extension of previous results, in that unlike other criteria of this type, no upper bound is placed on the derivative of the nonlinearity. An example is given in which this new criterion is used to obtain stability conditions for a non-linear sampled-data system. These conditions are less restrictive than those obtained using other stability criteria.

Graphical analysis of nonlinear sampled-data control-system dynamic behaviour

Based on previous work, the letter develops a graphical method for analysing the dynamic behaviour of nonlinear sampled-data control systems described by finite-difference equations.

Signal stabilization of nonlinear sampled-data control systems

Different methods are put forth for the analysis of stability and performance of nonlinear sampled-data systems, and many methods of stabilization are suggested. The injection of high-frequency oscillations at the input to the non-linear element is suggested as a possible method of stabilizing nonlinear sampled-data systems. Analytical and experimental investigations are carried out, and the advantages of signal stabilization over other methods are established.

Absolute stability of a class of nonlinear sampled-data systems

The present investigation studies the problem of absolute stability of a class of nonlinear sampled-data control systems with or without integrators in the loop. An absolute-stability criterion has been obtained by the second method of Lyapunov. The same stability criterion has been derived previously by the authors via the Popov approach. The criterion is shown to be a sufficient condition for the existence of a certain type of Lyapunov function which assures global-asymptotic stability of the class of systems under investigation. In contrast to previous results, the criterion does not place any restriction on the number of integrators in the loop. A systematic step-by-step method for applying the inequality is given, and an example illustrating the application of this frequency-domain inequality and a comparison with previous results are presented. The method is found to be versatile and more effective, and, in general, a better stability boundary can be obtained.

Adaptive sampling based on amplitude sensitivity

This paper presents the development of an algorithm for control of the sampling interval in discrete systems. The proposed strategy is based on the use of a sensitivity function which reflects the influence of discrete changes in hold circuit output on system response. The actual algorithm is hybrid, including both continuous computation and logical decisions. The method is illustrated by examples with a linear and a nonlinear sampled-data control system. The adaptive systems, where the sampling interval is variable, are shown to lead to significant reductions in the total number of samples required in an interval, as compared to equivalent systems with a fixed sampling rate.

Forced oscillations and suppression of oscillations in nonlinear sampled-data systems

In this paper, a procedure is developed to investigate and predict the properties of certain types of forced oscillations in a class of nonlinear sampled-data control systems. Use is made of the discrete describing function, and the results are checked on the IBM 7094 digital computer. A sinusoidal input signal with variable amplitude and phase is applied to a relay-type sampled-data control system. It is shown that the input signal has the effect of changing the shape of the critical regions of the negative-inverse of the discrete describing function. The amplitude, phase, and frequency of the forced oscillation can be computed from the intersection between the critical region and frequency loci of the linear portion of the system. Proper selections of amplitude, phases and frequency of the forcing sinusoidal signal permit the elimination and suppression of certain types of sustained oscillations in the system.

Stability and sensitivity of nonlinear sampled data systems

Publisher Summary This chapter discusses stability and sensitivity of nonlinear sampled data systems. Nonlinear sampled data control systems (SDS) are becoming increasingly applied in modern control engineering. Their application embraces process control system, radio locating systems, and digital computer control systems. In some cases, nonlinear SDS are used as models of physiological systems. Nonlinearities involved in SDS may be of two types: (1) those inherent in the system and because of limited power resources (saturation) and component inaccuracies, and (2) those intentionally introduced into the systems to improve its properties. The chapter highlights that every nonlinear SDS must be stable. It is essential that SDS be able to retain its properties at least under small variations of its parameters and of the external disturbing excitation. The chapter also discusses equations of nonlinear SDS and stability of the equilibrium position.

Analytical approaches to non-linear sampled-data control systems

The analysis of non-linear oscillations in sampled-data control systems is treated. The system considered here has the non-linear element in discrete signal portions. In the first half of the paper the strict analytical method is proposed. By using it, the periodic equilibrium states and their stability can be strictly analysed, without any assumption on the waveform, if the period of the oscillation is an integer multiple of the sampling period. In the second half, two different applications of describing function methods to the sampled-data systems are proposed. One is defined in the same way as in continuous systems, and it is accurately applicable to the sinusoidal oscillation of which the period is sufficiently larger than the sampling period. The other is defined by considering the phase relation of oscillations at sampling instants, and it is applicable when both frequencies are closely approached. Moreover, in some examples illustrated, the existence of ‘hard self-excitations’ in sampled-data systems is suggested.

Stability of nonlinear sampled-data control systems

With regard to the asymptotic stability in the large of nonlinear, autonomous sampled-data systems, the following conjecture has been quoted frequently in the literature. Namely, "if the lineafized system is stable for all points of the state space, the original nonlinear system is asymptotically stable in the large" (henceforth abbreviated a.s.i.l.). This paper shows by counter examples that the statement is not true in general. Both total and incremental linearizations are considered and in both cases the conjecture is false. Finally, a sufficient condition for a a.s.i.l, is given.

Analysis of Non-linear Sampled-data Control Systems by a Method of Linear Response Correction†

ABSTRACT The method of analysis outlined in this paper proceeds by developing an equivalent linear system having the same response to a given test input as would be obtained with the same input applied to the non-linear system. An analysis of the equivalent linear system then makes possible the evaluation of the system response The transformation from the non-linear system to its linear equivalent is made by forming a secondary input sequence, called a correction sequence, the response to which combines with the response to the test input to maintain the correct amplitude relationship between signals developed at the point of non-linearity In this paper the procedure is adapted to the Z-transform method of analysis and gives the exact response at sampling instants. No restriction is placed on the form of the non-linearity but the method is restricted in application to systems in which the non-linearity is directly associated with a data-hold circuit It is possible quickly to observe the effects of modification of the transfer characteristic of the non-linearity so that the method can form the basis of a procedure for developing improved systems by shaping of the non-linearity As a demonstration of the method, an example is given of the analysis of a particular problem.

Analysis of nonlinear sampled-data control systems -I

AN INTERESTING SIMILARITY exists between the mechanical sampling and zero-order hold function of sampled-data systems, and the mathematical "sampling" which is characteristic of numerical methods of analysis. This paper describes an investigation into this similarity. The results are numerical methods of analysis principally adapted to the transient solution of nonlinear sampled-data control systems. While numerical methods generally provide approximate solutions to nonlinear continuous systems, numerical methods applied to many nonlinear sampled-data systems provide exact solutions. Approximate solutions exist when exact solutions are not possible, i.e., for the more complex system configurations.

Analysis of nonlinear sampled-data control systems - II

AN INTERESTING SIMILARITY exists between the mechanical sampling and zero-order hold function of sampled-data systems, and the mathematical ?sampling? which is characteristic of numerical methods of analysis. This paper describes an investigation into this similarity. The results are numerical methods of analysis principally adapted to the transient solution of nonlinear sampled-data control systems. While numerical methods generally provide approximate solutions to nonlinear continuous systems, numerical methods applied to many nonlinear sampled-data systems provide exact solutions. Approximate solutions exist when exact solutions are not possible, i.e., for the more complex system configurations.